# Tagged Questions

Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

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### What are “classical groups”?

Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term classical group seems to have a fuzzier definition. Apparently it originates with ...
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### Origin of terms “flag”, “flag manifold”, “flag variety”?

These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably ...
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### What to do now that Lusztig's and James' conjectures have been shown to be false?

Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been ...
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### What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...
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### Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
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### Is the group of integer points on a finite-type group scheme over Z finitely presented?

Let $G$ be a group scheme of finite type over $\mathbf{Z}$. Must $G(\mathbf{Z})$ be finitely presented? (The question is inspired by a not yet successful attempt to answer a question of Brian ...
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I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own. I'm becoming increasingly fascinated by stuff ...
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### Is an algebraic space group always a scheme?

Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme? My feeling is that the answer is "yes" ...
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### Deformation theory of representations of an algebraic group

For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that the obstruction to deforming V as a representation of G is an element of H2(G,V&...
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### Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up. In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
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### A slick proof of the Bruhat Decomposition for GL_n(k)?

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the ...
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### Why is this not an algebraic space?

This question is related to the following question which I've just seen which was posted by Anton. His question is whether a algebraic space which is a group object is necessarily a group scheme, and ...
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### algebraic group G vs. algebraic stack BG

I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
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### Is there a “universal group object”? (answered: yes!)

I want to say that a group object in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One ...
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### When is fiber dimension upper semi-continuous?

Suppose $f\colon X \to Y$ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$. When is this function ...
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### Shuffle Hopf algebra: how to prove its properties in a slick way?

Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$. We define a $k$...
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### Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
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### Computing the Zariski closure of a subgroup of SL(n,Z)

Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is ...
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### Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...
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### Small-index subgroups of SL(3,Z)

I would like to know the smallest-index subgroups of ${\rm SL}(3,\mathbb{Z})$. The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$, along the bottom row. I could not figure out ...
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### Questions about the Bernstein center of a $p$-adic reductive group

Dear all, The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-...
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### How many ways can I factor a matrix (over $\mathbb{Z}$)?

Let $A$ be a fixed matrix in $M_2\mathbb{Z}$ with determinant $n \neq 0$. Question 1 How many ways can I write $A = XY$ for $X, Y \in M_2\mathbb{Z}$? The answer to this question is pretty clearly ...
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### Which p-adic algebraic groups are type I?

It was proved by Jacques Dixmier (Sur les représentations unitaires des groupes de Lie algébriques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820 , Zbl 0080....
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### Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with ...
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### Spin group as an automorphism group

Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with $p$...
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### Do semisimple algebraic groups always have faithful irreducible representations?

For simplicity, I will be talking only about connected groups over an algebraically closed field of characteristic zero. The basic theorem of affine algebraic groups is that they all admit faithful, ...
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### Quotients of Schemes by Free Group Actions

I've often seen people in seminars justify the existence of a quotient of a scheme by an algebraic group by remarking that the group action is free. However, I'm pretty sure they are also invoking ...
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### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense: Conjecture ...
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### Solving equations in SO(3) : an open problem by Jan Mycielski

I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, http://www.jstor.org/stable/...
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### How bad can $\pi_1$ of a linear group orbit be?

Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...
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### Comparing algebraic group orbits over big and small algebraically closed fields

For an affine algebraic group $G$ it's often convenient (and harmless) to work concretely over an algebraically closed field of definition $k$ while identifying $G$ with its group of rational points ...
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### Hopf Algebra Reference

I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf ...
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### Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer. I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order. Or equivalently, ...
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The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary, the result of historical accidents. In order to avoid repetitions, the four infinite families $A_\ell, B_\ell, C_\... 3answers 3k views ### A reference for geometric class field theory? The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ... 3answers 1k views ### Simple Tamagawa number calculations As is well known, Euler proved the Basel identity$\displaystyle\sum\limits_{i=0}^{\infty} \frac{1}{n^2} = \frac{{\pi}^2}{6}$. By far the most illuminating explanation of this fact that I've seen is ... 6answers 1k views ### How do I stop worrying about root systems and decomposition theorems (for reductive groups)? I apologize for this being a very very vague question. Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall ... 2answers 616 views ### Is the wonderful compactification of a spherical homogeneous variety always projective? Let$G/H$be a spherical homogeneous variety, where$G$is a complex semisimple group. Assume that the subgroup$H$is self-normalizing, i.e.,$\mathcal{N}_G(H)=H$. Then by results of Brion and Pauer ... 4answers 701 views ### Groups of matrices in which all elements have all eigenvalues equal in modulus I am writing a research article in which I need to use the following fact: if$G$is a subgroup of$GL_3(\mathbb{R})$which is irreducible in the sense that no proper nontrivial subspace of$\mathbb{R}...
Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$. 1. Some motivation A vector group is an ...
### Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?
Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...